Finite Group Databasebeta

← #27279817

Subgroups — #27279817

Order 13568.

694 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 139 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283054C211yesmaximal, characteristic
2#20274783C211yesmaximal, characteristic
3C2×#16810359C212yesmaximal
4#20272594C212yesmaximal
5C2×C53×#376C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7C2×C22⋊#88322C2212yes—
8#16810529C412yescomplemented
9#16810521C412yescomplemented
10#16810507—22no—
11#16810496—21no—
12#16810486—21no—
13#16810483C2211yescharacteristic
14#16810378—22no—
15#16810359—24no—
16C22×C42×C53—21no—
17C22×C42×C53C2211yescharacteristic
18C53×#376—24no—
19C2×#95×C53—21no—
20C2×#95×C53C2212yes—
21C22×Dic106C2×C411yescharacteristic
22C22×#88322—22no—
23#13133196—22no—
24#13133196—22no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2×C412yes—
27C2×#11719919—22no—
28C2×#11719919—21no—
29C2×C22⋊Dic53—21no—
30C2×C22⋊Dic53—21no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133124—22no—
34#13133124—22no—
35#13133124—22no—
36#13133124—21no—
37C23×C4×C53—22no—
38C23×C4×C53D411yescharacteristic
39C23×C4×C53D411yescharacteristic
40C23×C4×C53C2311yescharacteristic
41C2×C42×C53—28no—
42C2×C42×C53—24no—
43C2×C42×C53—22no—
44C2×C42×C53—22no—
45C2×C42×C53C2×C412yes—
46C2×C42×C53D412yes—
47#95×C53—24no—
48#95×C53—22no—
49#95×C53D414yes—
50C24×C53C2×D411yescharacteristic
51C22×C4×C53—21no—
52C22×C4×C53—21no—
53C22×C4×C53—24no—
54C22×C4×C53—28no—
55C22×C4×C53—21no—
56C22×C4×C53—24no—
57C22×C4×C53—21no—
58C22×C4×C53—22no—
59C22×C4×C53—22no—
60C22×C4×C53—42no—
61C22×C4×C53—44no—
62C22×C4×C53—42no—
63C22×C4×C53—42no—
64C22×C4×C53C22⋊C412yes—
65C22×C4×C53C22⋊C412yes—
66C22×C4×C53C22×C411yescharacteristic
67C22×C4×C53C2×D411yescharacteristic
68C22×C4×C53C2×D412yes—
69C22×C4×C53#4112yes—
70C22×Dic53—22no—
71C22×Dic53—21no—
72C22×Dic53—21no—
73C2×Dic106—22no—
74C2×Dic106—42no—
75C42×C53—216no—
76C42×C53—24no—
77C42×C53—24no—
78C22⋊Dic53—41no—
79C22⋊Dic53—41no—
80C22⋊Dic53—42no—
81C4⋊Dic53—22no—
82C4⋊Dic53—42no—
83#11719919—44no—
84#11719919—42no—
85C2×#88322—44no—
86C2×#88322—42no—
87C23×C53—22no—
88C23×C53—22no—
89C23×C53C2×C22⋊C411yescharacteristic
90C23×C53#12011yescharacteristic
91C23×C53#12311yescharacteristic
92C23×C53#13512yes—
93C23×C53#13712yes—
94C2×C4×C53—24no—
95C2×C4×C53—24no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—28no—
101C2×C4×C53—22no—
102C2×C4×C53—22no—
103C2×C4×C53—22no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—44no—
107C2×C4×C53—48no—
108C2×C4×C53—48no—
109C2×C4×C53—42no—
110C2×C4×C53—44no—
111C2×C4×C53—42no—
112C2×C4×C53#10414yes—
113C2×C4×C53C4×D412yes—
114C2×C4×C53#12112yes—
115C2×Dic53—41no—
116C2×Dic53—41no—
117C2×Dic53—42no—
118C2×Dic53—42no—
119C2×Dic53—42no—
120C2×Dic53—42no—
121C2×Dic53—42no—
122Dic106—82no—
123#88322—82no—
124#9145—531nomaximal, Sylow
125C22×C53—21no—
126C22×C53—21no—
127C22×C53—21no—
128C22×C53—21no—
129C22×C53—24no—
130C22×C53—24no—
131C22×C53#38611yescharacteristic
132C22×C53#41012yes—
133C22×C53#41611yescharacteristic
134C22×C53#41911yescharacteristic
135C22×C53C2×#10412yes—
136C22×C53#45212yes—
137C22×C53#48512yes—
138C4×C53—24no—
139C4×C53—22no—
140C4×C53—22no—
141C4×C53—48no—
142C4×C53—42no—
143Dic53—81no—
144Dic53—81no—
145Dic53—82no—
146#2855—531no—
147C2×#376D5311yescharacteristic
148#1817—531no—
149C2×#328—532no—
150#1294—532no—
151C2×C53—22no—
152C2×C53—22no—
153C2×C53#150012yes—
154C2×C53#150612yes—
155C2×C53#187311yescharacteristic
156C2×C53#189611yescharacteristic
157C2×C53#189911yescharacteristic
158C2×C22⋊Q8—531no—
159C22×C42—21no—
160C22×C42C2×D5311yescharacteristic
161C2×C22⋊C8—532no—
162#391—532no—
163#389—532no—
164#387—1062no—
165#385—1061no—
166#381—1061no—
167#380—531no—
168#376—24no—
169C2×#95—21no—
170C2×#95C2×D5312yes—
171#336—1062no—
172#328—1064no—
173C53#914511yescharacteristic, complemented, Sylow
174C22×Q8—531no—
175C23×C4—22no—
176C23×C4C22⋊D5311yescharacteristic
177C23×C4C22⋊D5311yescharacteristic
178C23×C4C22×D5311yescharacteristic
179C22×C8—1062no—
180C22⋊Q8—1062no—
181C22⋊Q8—1062no—
182C2×C4⋊C4—531no—
183C2×C4⋊C4—532no—
184C2×C4⋊C4—1061no—
185C2×C4⋊C4—1062no—
186C2×C22⋊C4—1061no—
187C2×C22⋊C4—1061no—
188C2×C42—24no—
189C2×C42—28no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C4×D5312yes—
193C2×C42D21212yes—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196#95—24no—
197#95—22no—
198#95—1061no—
199#95—1062no—
200#95—1062no—
201#95—1062no—
202#95C22⋊D5314yes—
203C24C2×C22⋊D5311yescharacteristic
204C2×Q8—1062no—
205C2×Q8—2122no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1062no—
220C22×C4—1061no—
221C22×C4—1061no—
222C22×C4#1171992112yes—
223C22×C4#1171992112yes—
224C22×C4C2×C4×D5311yescharacteristic
225C22×C4C2×D21211yescharacteristic
226C22×C4#1171994512yes—
227C22×C4C2×C22⋊D5312yes—
228C2×C8—2124no—
229C2×C8—2122no—
230C4⋊C4—1062no—
231C4⋊C4—2122no—
232C4⋊C4—2124no—
233C4⋊C4—2122no—
234C22⋊C4—2121no—
235C22⋊C4—2121no—
236C22⋊C4—2122no—
237C42—216no—
238C42—24no—
239C42—24no—
240C23—22no—
241C23—22no—
242C23C2×#1171992111yescharacteristic
243C23#1313318611yescharacteristic
244C23#1313320011yescharacteristic
245C23#1313323712yes—
246C23#1313324712yes—
247Q8—4242no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—28no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—48no—
261C2×C4—48no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—2121no—
267C2×C4—2121no—
268C2×C4—2122no—
269C2×C4—2122no—
270C2×C4—2122no—
271C2×C4—2122no—
272C2×C4—2122no—
273C2×C4#1313314514yes—
274C2×C4C4×C22⋊D5312yes—
275C2×C4#1313319112yes—
276C8—4242no—
277C22—24no—
278C22—24no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22#1681050211yescharacteristic
284C22#1681059612yes—
285C22#1681061711yescharacteristic
286C22#1681063111yescharacteristic
287C22C2×#1313314512yes—
288C22#1681078512yes—
289C22#1681096412yes—
290C4—24no—
291C4—22no—
292C4—22no—
293C4—48no—
294C4—42no—
295C4—4242no—
296C4—4241no—
297C4—4241no—
298C2—22no—
299C2—22no—
300C2#2027319912yes—
301C2#2027322612yes—
302C2#2027502711yescharacteristic
303C2#2027515811yescharacteristic
304C2#2027517311yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#376C2
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenterC2×#376D53
hypocenterC53#9145
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#376

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
8. #16810529C4semidirect
9. #16810521C4semidirect
173. C53#9145semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202830542
#202747832
C2×#168103592
#202725942
C2×C53×#3762
#914553